I think he is moreso saying that a measurement of the particle over a time interval less than infinity would lead to the creation of electron-positron pairs rendering the statement that a specific particle you are measuring had momentum $p$ meaningless.
It is still a big leap to go from this to $\hat{H}$ at $t=\pm \infty$ reduces to the free $H$. We can still use particles obeying the full $\hat{H}$ to measure the momentum of particle $A$ by just using very low energy particles of the full $H$ to collide with particle $A$ meaning no electron positron pairs will be produced